Keywords: dynamic stiffness method, free vibration, composite beams, geometrical and material coupling, Wittrick-Williams algorithm.

The free vibration analysis of composite beams [1,2,3] is an important and intense area of research activity, particularly because of its applications in aeronautical design [4,5]. Such an analysis is also used as a prerequisite to carry out aeroelastic or response analysis [6,7,8]. It is well recognised that the free vibration and response characteristics of composite beams can be very different from their metallic counter parts. This is because coupling between various modes of deformation that is not generally found in isotropic metals can occur in composites as a result of their anisotropic properties. From an aeroelastic point of view, particularly when designing composite wings this is significant. The particular coupling between the bending and torsional motion in a high aspect ratio aircraft wing, which can cause instability such as flutter, is a research topic of considerable importance [5,6,7,8].

The current paper is concerned with the dynamic stiffness formulation and free vibration analysis of composite beams which exhibit bending-torsion coupling. Such coupling arises from two principal sources. One of them stems from non-coincident shear centre and centroid of the cross-section [9]. This type of coupling is referred to as geometric coupling as it involves only the geometry of the cross-section. The nature of this bending-torsion coupling is inertial and as a consequence, the bending and torsional motions are generally uncoupled under static loading condition. The other type of coupling is confined to composite beams and it occurs due to anisotropic material properties (generally caused by fibre orientations). Clearly such type of coupling is possible under both static and dynamic loads. This particular form of coupling is referred to as material coupling because the coupling is dependent on the material properties only. Even for a doubly symmetric cross-section (for example, a solid rectangular or a box section), bending-torsion coupling can still occur in a composite beam due to ply orientations in the laminate [10]. As this particular coupling is due entirely to material properties, the geometry of the cross-section is not directly relevant. It is important to note that by splitting the coupling terms into geometric and material components provides a better understanding of and insight into the problem.

Within the above context the dynamic stiffness matrix of a bending-torsion coupled composite beam with effects of both geometric and material coupling is developed in order to investigate its free vibration characteristics. The governing differential equations of motion of the composite beam are derived using Hamilton's principle and the equations are solved analytically. The solution is used to obtain the dynamic stiffness matrix, which relates harmonically varying forces with harmonically varying displacements of the composite beam at its ends. Finally the resulting dynamic stiffness matrix is applied with particular reference to the Wittrick-Williams algorithm [11] to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn.

1

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2

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3

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4

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5

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6

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7

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8

Eslimy-Isfahany, S.H.R., and Banerjee, J.R., "Dynamic response of composite beams with application to aircraft wings", Journal of Aircraft, 34, 785-791, 1997. doi:10.2514/2.2244

9

Banerjee, J.R., "Coupled bending-torsional dynamic stiffness matrix for beam elements", International Journal for Numerical Methods in Engineering, 28, 1283-1298, 1989. doi:10.1002/nme.1620280605

10

Banerjee, J.R., and Williams, F.W., "Free vibration of composite beams - an exact method using symbolic computation", Journal of Aircraft, 32, 636-642, 1995. doi:10.2514/3.46767

11

Wittrick, W.H. and Williams, F.W., "A General algorithm for computing natural frequencies of elastic structures", Quarterly Journal of Mechanics and Applied Mathematics, 24, 263-284, 1971. doi:10.1093/qjmam/24.3.263

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